Abstract
The velocity decomposition of Saari ( Celest. Mech. 33, 299, 1984) is shown to capture the best possible effective potential of the general N-body problem. This improves the effective potential inherent to Sundman's inequality. Saari's approach is simplified by virtue of the inertia tensor and inertia ellipsoid. The inequality stronger than Sundman's obtained for the flat problem by Saari ( Celest. Mech. 40, 197, 1987) is not only extended to the general N-body problem but is also shown to be valid under more relaxed conditions. For the spatial three-body problem, we also obtain a more explicit expression of the inequality which was used to plot the Hill-type surfaces by Ge and Leng ( Celest. Mech. 53, 233, 1992). It is proved that Hill-type stability guarantees one of the hierarchical stability conditions, and prevents cross-over of orbits.
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